Here is the way I do an upgrade...works for me, but use at your own risk :-)
1. I don't delete any files. I create a new directory named something like oldsitebackup. Then I move all the existing files and directories into the oldsitebackup directory with the exception of the following:
1. Leave everything in wp-content in place except for blogs.php and index.php. I create a subdirectory in oldsitebackup called wp-content and then move those two files there. Also, if you are upgrading to 1.2.5, then delete (or move) the default plugins from your existing mu-plugings directory.
2. Leave wp-config.php and .htaccess in place (and leave favicon.ico if you created a custom favicon).
Move all other files to oldsitebackup. This will result in an in-place backup of all your source files.
Then upload the files from your upgrade with the exception of the following.
1. Don't upload the new wp-content directory, but do upload the blogs.php and index.php files that you moved out of your existing wp-content directory.
2. Upload all other directories and files with the exception of the ones you left in place.
3. If you have done any hacks to the core files (which I have done a lot), then apply them to the new files if needed.
Now you have your new files and directories in place and you can complete the upgrade process.
Be sure to back up your database before doing any upgrade, so you will be able to recover if you run into problems.
Steve